Isobaric Vapor–Liquid Equilibrium of the Mixture of Methyl Palmitate

Article pubs.acs.org/jced

Isobaric Vapor−Liquid Equilibrium of the Mixture of Methyl Palmitate and Methyl Stearate at 0.1 kPa, 1 kPa, 5 kPa, and 10 kPa Jun Hou,†,‡ Shimin Xu,†,‡ Hui Ding,*,†,‡ and Tao Sun†,‡ †

School of Chemical Engineering and Technology, Tianjin University, Tianjin, 300072 China National Engineering Research Center of Distillation Technology, Tianjin, 300072 China

‡

ABSTRACT: Isobaric vapor−liquid equilibrium (VLE) data of the mixture of methyl palmitate and methyl stearate at 0.1 kPa, 1 kPa, 5 kPa, and 10 kPa were measured with a modified Othmer still. Methyl palmitate and methyl stearate were the major components of the biodiesel. The experimental results were correlated with Wilson and nonrandom two-liquid (NRTL) activity coefficient models and were compared with a group contribution model (UNIFAC). The binary interaction parameters of the Wilson and NRTL models were fitted into the experimental data which were obtained at different pressures. The UNIFAC model did not predict the VLE for methyl palmitate and methyl stearate at reduced pressures well. The Wilson and NRTL models could well correlate with the experimental data at 1 kPa, 5 kPa, and 10 kPa. However, for the lower pressure 0.1 kPa, the Wilson model and NRTL model had worse correlatations with our experimental data than the higher pressures. These models could develop VLE calculations of the mixtures and provide guidelines for the separation of components by vacuum distillation.

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INTRODUCTION The energy crisis has become one of the most crucial issues in recent years. The unstable price of petroleum fuel in the world market and recent environmental concerns on gas emission during combustion have led to an intensive search for alternative energy sources that are not only renewable but also sustainable.1 Biodiesel refers to a vegetable oil or animal fat-based diesel fuel consisting of long-chain alkyl esters. Biodiesel, typically produced from reactions between lipids and alcohol, is a kind of clean, green, and renewable energy. Various methods such as direct use, blending, microemulsification, pyrolysis, and transesterification have been used for the production of biodiesel from vegetable oil.2,3 Among these methods, transesterification is a widely accepted technique and can be catalyzed by alkalis,4 acids,5 or enzymes.6 Besides, macroalgae as a new raw material for biodiesel production is promising and appears to have a cost advantage.7,8 The phase equilibrium data of fatty esters are very important in the design, simulation, and optimization of the production of biodiesel. Although lots of studies in biodiesel production have been reported, there are few articles focusing on measuring and predicting the vapor−liquid equilibrium (VLE) for components made of biodiesel due to the high boiling points at atmospheric pressure. Lockemann and Schluender9 measured the solid− liquid phase equilibria of binary mixtures of methyl myristate and methyl palmitate at ambient pressure. Rose and Scrohdt10 obtained the equilibrium data of a binary mixture of methyl oleate and methyl stearate by the total pressure technique and revealed a slight nonideality in the system. Rose and Supina11 measured vapor pressure and vapor−liquid equilibrium data for methyl esters of the common saturated normal fatty acids, but they only gave two VLE data of methyl palmitate and methyl stearate system at 4.0 kPa. Monick et al.12 measured VLE data of methyl palmitate and methyl stearate system at 0.53 kPa, and the © 2012 American Chemical Society

compositions of samples were determined by saponification values. Goodwin and Newsham13 measured the rate of thermal decomposition of methyl linoleate in the liquid phase and determined VLE of mixtures of methyl linoleate and methyl palmitate at 4.0 kPa. Ceriani and Meirelles14 proposed a group contribution method for the estimation of the vapor pressure of fatty compounds and gave good agreement with experimental data of fatty mixtures equilibrium with UNIFAC models. Akisawa et al.15 determined the VLE data of fatty acid ethyl esters using differential scanning calorimetry (DSC) and obtained the binary interaction parameters of the Wilson, nonrandom two-liquid (NRTL), and universal quasichemical activity coefficient (UNIQUAC) models. Considering the scarcity of the VLE data of the methyl esters and their importance to the biodiesel industry, methyl palmitate (C17H34O2) and methyl stearate (C19H38O2), the major components of biodiesel, would be studied in this study. Methyl palmitate can be used as an emulsifier, wetting agent, and stabilizer and can be a promising nucleus of anti-inflammatory and antifibrotic drugs.16 Methyl stearate can be used as a surfactant, lubricant, and so on. These two materials are important in life and industry. The VLE data of these two components at 0.1 kPa, 1 kPa, 5 kPa, and 10 kPa were determined in a modified Othmer still. Additionally, a group contribution model (UNIFAC) was applied to predict the VLE data, and the Wilson and NRTL models were correlated with the experimental data. The VLE data of the mixture of methyl palmitate and methyl stearate are essential for a better understanding of the thermodynamic behavior of such systems. Received: November 3, 2011 Accepted: August 30, 2012 Published: September 13, 2012 2632

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Article

Nitrogen was used as the carrier gas at a constant flow rate of 1 mL·min−1. The injector, detector, and oven temperatures were kept at 523.15 K, 533.15 K, and 473.15 K, respectively. Three analyses were made for each sample, and the averaged value was recorded.

EXPERIMENTAL SECTION Chemicals. The chemical reagents used were methyl myristate, methyl palmitate, and methyl stearate with a purity more than 99 %. They were all purchased from Shanghai Aoke Industrial Co., China. The purity of reagents were checked by gas chromatography (GC-2060, China), and the GC analysis did not show any observable impurities for the reagents, so all of the chemicals were used without further purification. Apparatus and Procedure. In this study, the VLE data was measured by a circulation VLE still (a modified Othmer still).17−19 The apparatus was shown in Figure 1. The internal volume of the

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RESULTS AND DISCUSSION To test the performance of equilibrium apparatus, the VLE data of the mixture of methyl myristate and methyl palmitate was measured at 4.0 kPa. Our experimental data were in good agreement with those reported by Rose and Supina11 and Akisawa.15 Table 1 shows the comparison between the literature data and Table 1. T−x Data of Methyl Myristate (1) + Methyl Palmitate (2) System at 4.0 kPaa x1

u(x)

T/K (present study)

u(T)

T/K (ref 11)

T/K (ref 15)

0.3460 0.4920 0.4960 0.7670

0.0015 0.0010 0.0010 0.0020

475.49 468.90 468.82 464.62

0.3 0.2 0.2 0.3

473.75 469.95 469.85 463.95

472.84 468.97 468.89 464.06

a

The deviations between the compositions of the gravimetrically composed samples and the temperatures indicated standard uncertainties u(q) in the quantity q as discussed in the text (0.95 level of confidence). The uncertainty of the pressure measurement was ur(p) = 0.01. The maximum expanded uncertainties of the temperature and composition measurements were assumed to be below 0.6 K and 0.005 mole fraction.

the data obtained by cubic spline interpolation of the data determined in our study. It was verified that the equilibrium apparatus was reliable in Figure 2.

Figure 1. Experimental setup for VLE measurement. 1, heating bar; 2, liquid-phase sampling port; 3, glycerol; 4, equilibrium chamber; 5, mercury thermometer; 6, condensing coil; 7, ground joint; 8, four-way pipe; 9, desiccator; 10, latex rubber tube; 11, buffer tank; 12, vacuum pump; 13, vapor-phase sampling port.

still was about 50 cm3 of which 40 cm3 was occupied by the liquid solution. The liquid was injected into the boiling chamber and vacuum the system by vacuum pump (Leybold D16 B, Germany). After the desired pressure was reached and stable, the liquid started to be heated. The vapor was condensed in the condensing coil and immediately returned to the equilibrium chamber through the vapor-phase sampling port. After the pressure fluctuation was maintained at less than ur(p) = 0.01 and the equilibrium temperature was kept constant at least 30 min, the equilibrium was assumed to be established. Samples of the vapor and liquid phase were taken from the sampling ports and then were dissolved by hexane and analyzed by GC. The temperature was measured with a mercury thermometer with the accuracy of ± 0.1 K. The pressure was measured with Pirani vacuum gauge with an accuracy of ± 1 % when pressures were 0.1 kPa and 1 kPa and with a digital manometer with an accuracy of ± 1 % when pressures were 5 kPa and 10 kPa. The compositions of the liquid and vapor phases were analyzed by GC. The GC was calibrated with standard solutions that were prepared gravimetrically by an electronic balance (FA2004N, uncertainty of ± 0.0001 g). The deviations between the compositions of the gravimetrically composed samples indicated standard uncertainties 0.001 of mole fraction xi and yi. The GC was equipped with a flame ionization (FID) detector. The GC response was treated with N-2000 chromatography station. The chromatographic column was a FFAP column (30 m × 0.45 mm × 2.55 um).

Figure 2. y1−x1 diagram of the methyl myristate and methyl palmitate system at 30 mmHg: ■, experimental values; △, data in ref 11; ○, data in ref 15.

Isobaric VLE of the mixture of methyl palmitate and methyl stearate was determined at 0.1 kPa, 1 kPa, 5 kPa, and 10 kPa. These data were in parallel tested three times, and the averaged data are listed in Table 2. When it reaches balance, the chemical potential of each component in different phases are equal, or their fugacities are equal. The basic thermodynamic relationships are as follows20,21 2633

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Table 2. Experimental VLE Data for the Methyl Palmitate (1) + Methyl Stearate (2) Systema x1

y1

0.0000 0.1298 0.2423 0.3369 0.4459 0.5703 0.6700 0.7811 0.8714 0.9621 1.0000

0.0000 0.5250 0.6134 0.6860 0.7550 0.8081 0.8617 0.9100 0.9420 0.9758 1.0000

0.0000 0.1631 0.2450 0.2980 0.4169 0.4973 0.5976 0.7236 0.8026 0.9123 1.0000

0.0000 0.4747 0.5524 0.6061 0.6531 0.6940 0.7401 0.8200 0.8547 0.9302 1.0000

T/K 0.1 kPa 427.50 417.65 415.05 414.27 413.95 412.13 411.17 410.88 409.35 408.66 408.30 5 kPa 512.50 499.53 498.03 497.96 495.83 494.24 493.58 492.98 491.88 491.18 490.80

u(x)

u(y)

u(T)

x1

y1

T/K

0.0005 0.0010 0.0020 0.0010 0.0030 0.0010 0.0030 0.0020 0.0025 0.0035 0.0010

0.0005 0.0010 0.0010 0.0035 0.0025 0.0025 0.0035 0.0010 0.0045 0.0015 0.0010

0.3 0.2 0.2 0.3 0.1 0.4 0.3 0.1 0.2 0.2 0.3

0.0000 0.0953 0.1616 0.2532 0.3341 0.4524 0.5926 0.6950 0.8368 0.9527 1.0000

0.0000 0.4155 0.5016 0.5617 0.6262 0.7160 0.7889 0.8400 0.9087 0.9745 1.0000

0.0005 0.0020 0.0015 0.0035 0.0025 0.0010 0.0015 0.0020 0.0015 0.0030 0.0010

0.0005 0.0030 0.0010 0.0025 0.0025 0.0020 0.0005 0.0030 0.0015 0.0015 0.0010

0.2 0.1 0.4 0.3 0.1 0.2 0.2 0.3 0.4 0.2 0.3

0.0000 0.1750 0.3233 0.4096 0.4781 0.5513 0.6276 0.6927 0.7744 0.8674 0.9462 1.0000

0.0000 0.3846 0.5787 0.6489 0.7042 0.7352 0.7823 0.8223 0.8592 0.9129 0.9611 1.0000

1 kPa 472.20 461.65 459.70 458.35 457.60 456.18 455.05 454.55 453.65 452.15 450.90 10 kPa 535.50 526.15 523.15 521.15 520.15 517.15 516.15 515.15 514.15 512.15 511.65 511.25

u(x)

u(y)

u(T)

0.0005 0.0015 0.0025 0.0020 0.0020 0.0010 0.0035 0.0040 0.0015 0.0020 0.0010

0.0005 0.0025 0.0010 0.0020 0.0045 0.0030 0.0020 0.0015 0.0030 0.0010 0.0010

0.2 0.1 0.4 0.3 0.1 0.2 0.2 0.3 0.4 0.2 0.3

0.0005 0.0015 0.0015 0.0025 0.0015 0.0035 0.0015 0.0035 0.0025 0.0020 0.0030 0.0010

0.0005 0.0025 0.0015 0.0015 0.0030 0.0020 0.0020 0.0030 0.0015 0.0040 0.0010 0.0010

0.4 0.2 0.2 0.3 0.1 0.1 0.3 0.1 0.2 0.2 0.3 0.2

a The deviations between the compositions of the gravimetrically composed samples and the temperatures indicated standard uncertainties u(q) in the quantity q as discussed in the text (0.95 level of confidence). The uncertainty of the pressure measurements was ur(p) = 0.01. The maximum expanded uncertainties of the temperature and composition measurements were assumed to be below 0.8 K and 0.01 mole fraction.

⎛ V l(P − P s) ⎞ i ΦiPyi = ΦisPisγixi exp⎜⎜ i ⎟⎟ RT ⎠ ⎝

Table 3. Antoine Constants for the Pure Components Antoine constants

(1)

where yi and xi are the mole fractions of component i in the vapor and liquid phases, respectively; P and T are the total pressure and the temperature in the equilibrium system, respectively; Φi is the fugacity coefficient of component i in the vapor phase; γi is the activity coefficient of component i in the liquid phase; Φsi is the fugacity coefficient of the pure saturated vapor of component i; Psi is the saturated vapor pressure of component i at temperature T; Vli is the molar volume of pure liquid. In the expression, the vapor could be regarded as ideal gas at reduced pressure, thus Φi and Φsi are equal to 1;22 the term exp((Vli(P − Psi ))/RT) is also approximately equal to 1 at low pressure.23 Therefore, eq 1 can be written as

Pyi = γiPisxi

component

A

B

C

pressure range/ mmHg

methyl palmitate

7.53530 7.17574 7.45930 8.74432 5.42311

2285.800 2017.660 2313.600 3239.351 1023.847

166.000 142.557 152.000 212.867 25.048

0.1 to 6a 20 to 200b 0.1 to 6a 0.82 to 13c 20 to 200b

methyl stearate

a

Taken from ref 24. bTaken from ref 11. cTaken from ref 10.

where γiC is the entropic contribution to the nonideality of the mixture and γiR is the residual or energy-related contribution. The combinatorial part is obtained by the following relations

(2)

ln γi C = ln

The saturated vapor pressure could be calculated by the Antoine equation.10,11,24 log10(Pis/mmHg) = Ai −

Bi (T /K − 273.15 + Ci)

li = 5(ri − qi) − (ri − 1) (3)

θi = qixi /(∑ qjxj) j

where Ai, Bi, and Ci are three parameters for the Antoine equation, which are listed in Table 3. The activity coefficient of component i, γi, was calculated by the UNIFAC, Wilson, and NRTL models. In the UNIFAC model,25 γi was calculated by C

ln γi = ln γi + ln γi

R

N ⎛ θ⎞ Φi Φ + 5qi⎜ln i ⎟ + li − i (∑ xjl j) xi xi j = 1 ⎝ Φi ⎠

(5) (6)

(7)

Φi = rx i i /(∑ rjxj) (8)

j m

qi =

(4)

∑ υk(i)Q k k=1

2634

(9)

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m

∑ υk(i)R k

ri =

Table 6. Correlated Binary Interaction Parameters and RootMean-Square Deviations between the Experimental Data and Calculated Values for the Binary System with Different Models

(10)

k=1

where ri is the volume parameter for component i; qi is the area parameter for component i; υ(i) k is the number of groups of type k in component i; Rk is the volume parameter for group k; and Qk is the area parameter for group k. For the residual term, the equation is

methyl palmitate (1) + methyl stearate (2)

ln γi =

∑

υk(i)(ln Γk

(i)

− ln Γk )

(11)

k=1

where Γk is the group residual activity coefficient; Γ(i) k is the residual activity coefficient of group k in a reference solution containing only components of type i. The group activity coefficient Γk is found from ⎡ m ln Γk = Q k ⎢1 − ln(∑ θj̅φjk ) − ⎢ j=1 ⎣

which also holds for ln Γ(i) k . θ̅j

⎞⎤ ⎛ θj̅φ kj ⎟⎥ ⎜ ∑⎜ m ∑ θ ̅ φ ⎟⎥ j = 1 ⎝ n = 1 n nj ⎠⎦ m

(12)

is the area fraction of group j, which is

defined as θj̅ =

a

σTb/K

σy1b

−4.5723

2.09 1.96 1.91

0.0792 0.0377 0.0427

−0.9173

3.88 1.11 1.11

0.0796 0.0283 0.0295

−0.1696

5.94 0.80 0.92

0.0879 0.0154 0.0142

−1.2839

2.35 0.87 0.88

0.0394 0.0159 0.0160

A21a 0.1 kPa

m R

A12a

model

Q jXj m ∑n = 1 Q nxn

a

(13)

b

UNIFAC Wilson NRTL

0.1571 948.1

1.5833 738.6 1 kPa

UNIFAC Wilson NRTL

0.2388 2687.8

1.0697 158.2 5 kPa

UNIFAC Wilson NRTL

0.4249 3913.9

0.7519 −290.9 10 kPa

UNIFAC Wilson NRTL

1.3207 −513.4

0.4330 1656.8

Wilson, Aij = Λij; NRTL, Aij = gij−gjj, a = aij = aji. n

n

where Xj is the mole fraction of group j in the mixture. The group interaction parameter φjk is given by ⎛ ajk ⎞ φjk = exp⎜ − ⎟ ⎝ T⎠ (14)

σT =

∑i = 1 (Tical − Tiexp)2 n

;

σy1 =

∑i = 1 (y1,cali − y1,exp )2 i n

where ajk is the group interaction parameter. All of the group volume Rk and area Qk parameters and the group interaction parameters ajk are listed in Tables 4 and 5, respectively. Table 4. Group Volume Rk and Area Qk Parameters group

group number k

volume parameter Rk

area parameter Qk

−CH3 −CH2− −CH2COO−

1 2 3

0.9011 0.6744 1.6764

0.8480 0.5400 1.4200

Table 5. Group Interaction Parameter ajk between j and k Groups group

−CH3

−CH2−

−CH2COO−

−CH3 −CH2− −CH2COO−

0 0 114.8

0 0 114.8

232.1 232.1 0

Figure 3. T−x1−y1 diagram of the methyl palmitate (1) + methyl stearate (2) system at 0.1 kPa: ●, ○, experimental data in this study; , correlated results by the Wilson model; ---, correlated results by the NRTL model; ···, predicted results by the UNIFAC model.

In the Wilson model,26 γi was calculated by N

N

ln γi = 1 − ln(∑ Λijxj) − j=1

Λkixk N k = 1 ∑ j = 1 Λkjxj

∑

τji =

(15)

and in the NRTL model, γi was calculated by

gji − gii (17)

RT

27

N

ln γi =

∑ j = 1 τjiGjixj N

∑k Gkjxk

N

+

∑ j=1

Gji = exp( −αjiτji)

xjGij

Values of the interaction parameters Λij in the Wilson model and gji − gii and αji in the NRTL model were obtained by regression of the experimental VLE data. Naik et al.28 summarized the consistency of isobaric binary VLE data for methyl esters of fatty acids, such as the visual method and the composition-resolution method. However, the

N

∑k = 1 Gkjxk

n ⎛ ∑k = 1 xkτkjGkj ⎞ ⎜ ⎟ × ⎜τij − N ⎟ ∑ G x ⎝ k = 1 kj k ⎠

(18)

(16) 2635

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Figure 6. T−x1−y1 diagram of the methyl palmitate (1) + methyl stearate (2) system at 10 kPa: ●, ○, experimental data in this study; , correlated results by the Wilson model; ---, correlated results by the NRTL model; ···, predicted results by the UNIFAC model.

Figure 4. T−x1−y1 diagram of the methyl palmitate (1) + methyl stearate (2) system at 1 kPa: ●, ○, experimental data in this study; , correlated results by the Wilson model; ---, correlated results by the NRTL model; ···, predicted results by the UNIFAC model.

kPa, 1 kPa, 5 kPa, and 10 kPa were 5.57, −2.75, 7.77, and 7.47, respectively. This indicated that the experimental data were thermodynamically consistent. The procedure was simulated by the software of Matlab, and the VLE data were calculated. Values of the interaction parameters Λij in the Wilson model and gji − gii and αji in the NRTL model were determined by minimizing the following objective function F:30 2

F=

n

∑ ∑ (yical k=1 i=1

Herington29 method was the general method used to check the thermodynamic consistency. x =1

D = 100·

2

γ1

1

γ2

∫x = 0 ln J = 150·

γ

∫x =1 0 ln γ1 dx1 x1= 1

Tmax − Tmin Tmin

dx1

(21)

where n is the number of experimental data. The minimization method used was the nonlinear least-squares method. The regression binary intersection parameters of each model and the root-mean-squared deviations (rmsd) of the vapor mole fraction and temperature between the experimental data and the calculated values were listed in Table 6. The experimental VLE data and the simulated results at 0.1 kPa, 1 kPa, 5 kPa, and 10 kPa are plotted in Figures 3 to 6, respectively. The absolute deviations of the vapor composition (Δy1) and the temperature (ΔT) between three model values and experimental data are listed in Tables 7 and 8, respectively. It could be seen that the results calculated by UNIFAC model showed the largest deviation for methyl palmitate and methyl stearate system at all studied pressures. The average deviations of Δy1 and ΔT between the UNIFAC model values and the experimental data were 0.07 and 5 K, respectively. Therefore, UNIFAC model was not suitable for predicting the VLE of the mixture of methyl palmitate and methyl stearate at reduced pressures. Both Wilson and NRTL models were better correlated with the experimental data than the UNIFAC model. The average deviation of Δy1 and ΔT between the correlated model values and experimental data were all less than 0.014 and 0.81 K at 5 kPa and 10 kPa, respectively. However, the results calculated by the Wilson and NRTL models at 0.1 kPa and 1 kPa seemed worse than those at 5 kPa and 10 kPa. At the pressure of 1 kPa, the average

Figure 5. T−x1−y1 diagram of the methyl palmitate (1) + methyl stearate (2) system at 5 kPa: ●, ○, experimental data in this study; , correlated results by the Wilson model; ---, correlated results by the NRTL model; ···, predicted results by the UNIFAC model.

1

− yiexp )2

(19)

(20)

where Tmax and Tmin are the maximum and minimum boiling temperatures in the studied systems, respectively. Herington suggested that if D − J < 10, the experimental points are considered to be thermodynamically consistent. The values of D − J for the systems of methyl palmitate and methyl stearate at 0.1 2636

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Table 7. Comparison of Three Models with Experimental x1−y1 Data at Different Vacuum Pressures experiment data

UNIFAC model

x1

yexp 1

ycal 1

Δy1a

0.1298 0.2423 0.3369 0.4459 0.5703 0.6700 0.7811 0.8714 0.9621 ADb

0.5250 0.6134 0.6860 0.7550 0.8081 0.8617 0.9100 0.9420 0.9758

0.3232 0.5106 0.6271 0.7300 0.8191 0.8751 0.9257 0.9597 0.9890

0.2018 0.1028 0.0589 0.0250 0.0110 0.0134 0.0157 0.0177 0.0132 0.0511

0.0953 0.1616 0.2532 0.3341 0.4524 0.5926 0.6950 0.8368 0.9527 AD

0.4155 0.5016 0.5617 0.6262 0.7160 0.7889 0.8400 0.9087 0.9745

0.2383 0.3648 0.5036 0.6011 0.7138 0.8154 0.8741 0.9401 0.9840

0.1772 0.1368 0.0581 0.0251 0.0022 0.0265 0.0341 0.0314 0.0095 0.0557

0.1631 0.2450 0.2980 0.4169 0.4973 0.5976 0.7236 0.8026 0.9123 AD

0.4747 0.5524 0.6061 0.6531 0.6940 0.7401 0.8200 0.8547 0.9302

0.3008 0.4189 0.4863 0.6166 0.6913 0.7721 0.8577 0.9040 0.9604

0.1739 0.1335 0.1198 0.0365 0.0027 0.0320 0.0377 0.0493 0.0302 0.0684

0.1750 0.3233 0.4096 0.4781 0.5513 0.6276 0.6927 0.7744 0.8674 0.9462 AD

0.3846 0.5787 0.6489 0.7042 0.7352 0.7823 0.8223 0.8592 0.9129 0.9611

0.3129 0.5073 0.5999 0.6649 0.7274 0.7858 0.8311 0.8826 0.9349 0.9748

0.0717 0.0714 0.0490 0.0393 0.0078 0.0035 0.0088 0.0234 0.0220 0.0137 0.0311

Wilson model

NRTL model

ycal 1

Δy1

ycal 1

Δy1

0.4501 0.5655 0.6387 0.7126 0.7884 0.8440 0.9011 0.9440 0.9841

0.0749 0.0479 0.0473 0.0424 0.0197 0.0177 0.0089 0.0020 0.0083 0.0299

0.4316 0.5573 0.6368 0.7150 0.7928 0.8483 0.9042 0.9457 0.9845

0.0934 0.0561 0.0492 0.0400 0.0153 0.0134 0.0058 0.0037 0.0087 0.0317

0.3922 0.4820 0.5575 0.6074 0.6714 0.7452 0.8017 0.8870 0.9653

0.0233 0.0196 0.0042 0.0188 0.0446 0.0437 0.0383 0.0217 0.0092 0.0248

0.3898 0.4839 0.5593 0.6075 0.6693 0.7425 0.7998 0.8868 0.9656

0.0257 0.0177 0.0024 0.0187 0.0467 0.0464 0.0402 0.0219 0.0089 0.0254

0.4579 0.5389 0.5774 0.6456 0.6852 0.7332 0.7979 0.8442 0.9215

0.0168 0.0135 0.0287 0.0075 0.0088 0.0069 0.0221 0.0105 0.0087 0.0137

0.4578 0.5432 0.5828 0.6504 0.6884 0.7341 0.7964 0.8420 0.9196

0.0169 0.0092 0.0233 0.0027 0.0056 0.0060 0.0236 0.0127 0.0106 0.0123

0.3640 0.5474 0.6268 0.6805 0.7312 0.7787 0.8164 0.8617 0.9138 0.9620

0.0206 0.0313 0.0221 0.0237 0.0040 0.0036 0.0059 0.0025 0.0009 0.0009 0.0116

0.3625 0.5471 0.6270 0.6809 0.7315 0.7789 0.8164 0.8614 0.9134 0.9618

0.0221 0.0315 0.0219 0.0233 0.0037 0.0034 0.0059 0.0022 0.0005 0.0007 0.0115

0.1 kPa

1 kPa

5 kPa

10 kPa

a

Δy1, i = |y1,cali − y1,exp | i b

n

AD =

∑i = 1 |y1,cali − y1,exp | i n

deviation of Δy1 and ΔT between the correlated model values and experimental data were about 0.025 and 1.10 K, respectively. At the pressure of 0.1 kPa, the average deviation of Δy1 and ΔT between the correlated model values and experimental data were about 0.032 and 1.70 K, respectively.

With the pressure decreasing, the correlated result continued to get worse. From the view of industrial application, especially for vacuum distillation separation, the Wilson and NRTL models could be used to calculate the VLE of the binary system. 2637

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Table 8. Comparison of Three Models with Experimental T−x Data at Different Vacuum Pressures experiment data

UNIFAC model

Wilson model

x1

Texp/K

Tcal/K

ΔTa/K

0.1298 0.2423 0.3369 0.4459 0.5703 0.6700 0.7811 0.8714 0.9621 ADb

417.65 415.05 414.27 413.95 412.13 411.17 410.88 409.35 408.66

421.87 418.75 416.47 414.15 412.82 411.14 409.45 409.19 408.02

4.22 3.70 2.20 0.20 0.69 0.03 1.43 0.16 0.64 1.47

0.0953 0.1616 0.2532 0.3341 0.4524 0.5926 0.6950 0.8368 0.9527 AD

461.65 459.70 458.35 457.60 456.18 455.05 454.55 453.65 452.15

468.06 465.73 462.86 460.62 457.72 454.73 452.79 450.38 448.60

6.41 6.03 4.51 3.02 1.54 0.32 1.76 3.27 3.55 3.38

0.1631 0.2450 0.2980 0.4169 0.4973 0.5976 0.7236 0.8026 0.9123 AD

499.53 498.03 497.96 495.83 494.24 493.58 492.98 491.88 491.18

508.82 506.48 505.06 502.10 500.26 498.10 495.61 494.15 492.23

9.29 8.45 7.10 6.27 6.02 4.52 2.63 2.27 1.05 5.29

0.1750 0.3233 0.4096 0.4781 0.5513 0.6276 0.6927 0.7744 0.8674 0.9462 AD

526.15 523.15 521.15 520.15 517.15 516.15 515.15 514.15 512.15 511.65

530.36 525.89 523.57 521.85 520.12 518.41 517.02 515.37 513.60 512.18

4.21 2.74 2.42 1.70 2.97 2.26 1.87 1.22 1.45 0.52 2.14

NRTL model

Tcal/K

ΔT/K

Tcal/K

ΔT/K

417.71 415.19 413.59 411.99 410.34 409.12 407.85 406.87 405.93

0.06 0.14 0.68 1.96 1.79 2.05 3.03 2.48 2.73 1.66

418.34 415.67 413.95 412.24 410.49 409.22 407.90 406.89 405.93

0.69 0.62 0.32 1.71 1.64 1.95 2.98 2.46 2.73 1.68

462.62 460.13 458.06 456.77 455.28 453.84 452.93 451.84 451.09

0.97 0.43 0.29 0.83 0.90 1.21 1.62 1.81 1.06 1.01

462.76 460.12 458.01 456.75 455.30 453.88 452.97 451.87 451.10

1.11 0.42 0.34 0.85 0.88 1.17 1.58 1.78 1.05 1.02

500.72 498.08 496.87 494.88 493.88 492.87 491.86 491.37 490.91

1.19 0.05 1.09 0.95 0.36 0.71 1.12 0.51 0.27 0.69

500.85 498.03 496.74 494.69 493.69 492.70 491.72 491.26 490.85

1.32 0.00 1.22 1.14 0.55 0.88 1.26 0.62 0.33 0.81

527.57 522.39 520.06 518.47 516.99 515.65 514.64 513.53 512.46 511.68

1.42 0.76 1.09 1.68 0.16 0.50 0.51 0.62 0.31 0.03 0.71

527.67 522.45 520.08 518.47 516.98 515.63 514.62 513.51 512.44 511.68

1.52 0.70 1.07 1.68 0.17 0.52 0.53 0.64 0.29 0.03 0.71

0.1 kPa

1 kPa

5 kPa

10 kPa

a

ΔTi = |Tical − Tiexp| b

n

■

AD =

∑i = 1 |Tical − Tiexp| n

CONCLUSIONS The isobaric VLE data of the mixture of methyl palmitate and methyl stearate were determined at 0.1 kPa, 1 kPa, 5 kPa, and 10 kPa. The experimental data were checked with the Herington method, which showed good thermodynamic consistency. The UNIFAC model was used to predict the VLE data and had large deviations from experimental data at all studied pressures. The

Wilson and NRTL activity coefficient models could well correlate experimental data at 1 kPa, 5 kPa, and 10 kPa but were not good at 0.1 kPa. The results of this study did not only provide experimental VLE data of methyl palmitate and methyl stearate at different reduced pressures but also try to find the optimal thermodynamic models to develop the VLE calculations of the mixtures. 2638

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(20) Smith, J. M.; Van Ness, H. C.; Abbott, M. M. Introduction to Chemical Engineering Thermodynamics, 6th ed.; McGraw-Hill: New York, 2001. (21) Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria, 3rd ed.; Prentice Hall PTR: Upper Saddle River, NJ, 1998. (22) Chen, Z. X.; Gu, F. Y.; Hu, W. M. Chemical Thermodynamics Program; Chemical Industry Press: Beijing, 1993. (23) Li, Q.; Zhang, J.; Lei, Z.; Zhu, J.; Xing, F. Isobaric Vapor−Liquid Equilibrium for Ethyl Acetate + Ethanol + 1-Ethyl-3-Methylimidazolium Tetrafluoroborate. J. Chem. Eng. Data 2009, 54, 193−197. (24) Scott, T. A.; Macmillan, D.; Melvin, E. H. Vapor Pressures and Distillation of Methyl Esters of Some Fatty Acids. Ind. Eng. Chem. 1952, 44, 172−175. (25) Fredenslund, A.; Jones, R. L.; Prausnitz, J. M. GroupContribution Estimation of Activity Coefficients in Nonideal Liquid Mixtures. AIChE J. 1975, 21, 1086−1099. (26) Wilson, G. M. Vapor−Liquid Equilibrium. Xi. A New Expression for the Excess Free Energy of Mixing. J. Am. Chem. Soc. 1964, 86, 127− 130. (27) Renon, H.; Prausnitz, J. M. Local Compositions in Thermodynamic Excess Functions for Liquid Mixtures. AIChE. J. 1968, 14, 135− 144. (28) Naik, H. A.; Husain, A.; Chari, K. S. Consistency of Isobaric Binary Vapour−Liquid Equilibrium Data: Methyl Esters of Fatty Acids. Indian J. Technol. 1964, 2, 255−258. (29) Herington, E. Tests for the Consistency of Experimental Isobaric Vapor−Liquid Equilibrium Data. J. Inst. Petrol. 1951, 37, 457−470. (30) Yu, G. B.; Guo, K. H.; Sun, B. Vapor−Liquid Equilibrium Measurements for the Nitrogen + Tetrafluoromethane System over a Temperature Range of (134.27 to 204.85) K. J. Chem. Eng. Data 2009, 54, 2281−2284.

AUTHOR INFORMATION

Corresponding Author

*Tel.: +86-27405745. Fax: +86-27404705. E-mail: dinghui@tju. edu.cn. Funding

This work was financially supported by a grant from the Ph.D. Programs Foundation of Ministry of Education of China (No. 20090032120080). Notes

The authors declare no competing financial interest.

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